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A072474 Sum of next n squares.

%I A072474 #49 Jan 07 2025 08:13:15
%S A072474 1,13,77,294,855,2071,4403,8492,15189,25585,41041,63218,94107,136059,
%T A072474 191815,264536,357833,475797,623029,804670,1026431,1294623,1616187,
%U A072474 1998724,2450525,2980601,3598713,4315402,5142019,6090755,7174671,8407728,9804817,11381789,13155485
%N A072474 Sum of next n squares.
%H A072474 Kelvin Voskuijl, <a href="/A072474/b072474.txt">Table of n, a(n) for n = 1..10000</a>
%H A072474 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,20,-15,6,-1).
%F A072474 a(n) = k*(k+1)*(2*k+1)/6 - r*(r+1)*(2*r+1)/6, where k = n*(n+1)/2 and r = n*(n-1)/2.
%F A072474 a(n) = A000330(n*(n+1)/2) - A000330(n*(n-1)/2).
%F A072474 a(n) = (n/12)*(3*n^2 + 1)*(n^2 + 2). - _Benoit Cloitre_, Jun 26 2002
%F A072474 G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^6. - _Colin Barker_, Mar 23 2012
%F A072474 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n > 6. - _Jinyuan Wang_, May 25 2020
%F A072474 E.g.f.: exp(x)*x*(12 + 66*x + 82*x^2 + 30*x^3 + 3*x^4)/12. - _Stefano Spezia_, May 14 2024
%e A072474 a(1) = 1^2 = 1;
%e A072474 a(2) = 2^2 + 3^2 = 13;
%e A072474 a(3) = 4^2 + 5^2 + 6^2 = 77.
%t A072474 Table[Sum[ i^2, {i, n(n - 1)/2 + 1, n(n + 1)/2}], {n, 1, 35}]
%o A072474 (PARI) a(n) = n*(3*n^2+1)*(n^2+2)/12
%o A072474 (Magma) [n*(3*n^2+1)*(n^2+2)/12: n in [1..35]]; // _Vincenzo Librandi_, Dec 31 2024
%Y A072474 Cf. A000290, A000330, A072475, A050410.
%Y A072474 Cf. A006003 (for natural numbers), A260513 (for triangular numbers), A372583 (for pentagonal numbers), A372751 (for hexagonal numbers), A075664 (for cubes).
%K A072474 nonn,easy
%O A072474 1,2
%A A072474 _Amarnath Murthy_, Jun 20 2002
%E A072474 Edited by _Robert G. Wilson v_, Jun 21 2002